Walrasian Equilibrium with Gross Substitutes

Faruk Gul Princeton University and Ennio Stacchetti University of Michigan

Version: February 8, 1999

This research was supported in part by the National Science Foundation. Gul also thanks the Alfred P. Sloan Foundation for their support. We are grateful to Vincent Crawford who provided valuable comments and pointed out to us the connection between Kelso and Crawford [5] and our work, to John Geanakoplos for suggesting a weakening of our main assumption, and to Don Brown for valuable comments.

1. Introduction In this paper we study the problem of efficient production and allocation of indivisible objects among a set of consumers. We assume that the agents’ preferences depend on the bundle of objects and the quantity of money they consume. Furthermore, we assume that preferences are quasilinear in money, and that agents have a large endowment of money. With indivisibilities, it is well-known that many familiar properties of the utility functions fail to ensure existence. In their striking analysis of the matching problem, Kelso and Crawford [5] introduce the gross substitutes (GS) condition which ensures the nonemptiness of the core. We propose two new conditions, and show that with quasilinearity they are equivalent to the (GS) condition of Kelso and Crawford. The simplest example of (GS) preferences are unit demand preferences. A unit demand preference is such that the agent can enjoy at most one object. We prove that the set of (GS) preferences is the largest set containing unit demand preferences for which an existence theorem can be established. Thus, we prove a “converse” to Kelso and Crawford’s existence result; in a sense, the (GS) condition is necessary to ensure existence of a Walrasian equilibrium. With quasilinear preferences, there is a representative consumer whose demand function coincides with society’s aggregate demand. When the (GS) condition is satisfied, the smallest Walrasian price and the largest Walrasian price of each object can be interpreted as shadow prices. The largest Walrasian price of any object α represents the decrease in total utility of an efficient allocation that would result if this object were removed from the aggregate endowment. Similarly, the smallest Walrasian price represents the amount of increase associated with an efficient allocation if a second copy (i.e. a perfect substitute) of this object were added to the economy. Consequently, we show that the representative consumer’s utility function satisfies submodularity whenever the utility functions of the individual agents satisfy the (GS) condition. In Section 5 we compare Walrasian prices with Vickrey-Clarke-Groves payments. We prove that for any profile of preferences, the equilibrium payment of any agent in the VCG mechanism is less than or equal to the value of the allocation he receives at the smallest Walrasian prices. Therefore, the total revenue raised by the VCG mechanism is less than or equal to the value of the aggregate endowment at the smallest Walrasian prices. We show that these inequalities may be strict. However, the two inequalities are in fact equalities if the initial economy satisfies the (GS) condition and is replicated m + 1 times (where m is the number of objects in the initial economy). In Section 6 we generalize the model to include production. Kelso and Crawford’s analysis of the core of a matching problem plays a central role in our work. Their framework is more general than ours. In particular, they do not impose quasilinearity. We rely on their paper for existence of a competitive equilibrium and utilize quasilinearity to prove additional results. We compare our results to theirs throughout the paper. A different approach to the existence problem is provided in Bikhchandani and Mamer [2]. They construct a related economy with quasilinear preferences without indivisibilities. The total surplus attainable in this economy is no less than the total surplus attainable in the economy with indivisibilities. Their main theorem proves that equilibrium in the economy with indivisibilites exists if and only if the maximal attainable surplus 1

is equal to the maximal attainable surplus in the corresponding economy with no indivisibilities. They use this result to identify various sufficient conditions for existence with indivisibilities. The necessary and sufficient condition for their main theorem described above suggests the following alternative approach for proving existence of Walrasian equilibrium. One could verify directly that an economy satisfying the (GS) condition and its divisible analog defined in Bickhchandani and Mamer [2] yield the same surplus. 2. Preferences In this section we study properties of the consumers’ preferences. We confine attention to preferences that are quasilinear in money (that is, to additively separable utility functions), and study conditions on the preferences over bundles of objects. Ω = {ω1 , . . . , ωm } is the set of objects in the economy. A bundle is any subset B of Ω; the set of all bundles is 2Ω := {B | B ⊂ Ω}. A price vector p ∈ Rm + includes a price for each object in Ω. Definition: A map u : 2Ω → R is called a utility function on Ω. A utility function assigns a value to each bundle of Ω. With each utility function u we associate the net utility function v : 2Ω × Rm + → R, which is defined by v(A, p) := u(A) − < p, A >, where

< p, A > :=

X

pa

a∈A

(and by convention, < p, ∅ > := 0). Definition: A utility function u : 2Ω → R (i) is monotone if for all A ⊂ B ⊂ Ω, u(A) ≤ u(B). (ii) is submodular if for all A, B ⊂ Ω, u(A) + u(B) ≥ u(A ∪ B) + u(A ∩ B). (iii) has decreasing marginal returns if for all A ⊂ B ⊂ Ω and a ∈ A, u(B) − u(B\{a}) ≤ u(A) − u(A\{a}). If u(∅) = 0 and u is monotone, then u(A) ≥ 0 for all A ⊂ Ω. In what follows, without loss of generality, we normalize every utility function u so that u(∅) = 0. Conditions (i) – (iii), as well as the equivalence of conditions (ii) and (iii), are well known in the literature. Likewise, one can establish the equivalence between supermodularity (or convexity), which is obtained by reversing the inequality in the definition of submodularity, and increasing marginal returns (which is obtained by reversing the inequality in the definition of decreasing marginal returns). 2

Lemma 1: u is submodular iff u has decreasing marginal returns. The proof of this lemma can be found, for example, in Moulin [10]. Rm +

Definition: For any utility function u : 2Ω → R, its demand correspondence D : → 2Ω is defined by D(p) := { A ⊂ Ω | v(A, p) ≥ v(B, p) for all B ⊂ Ω },

p ∈ Rm +.

Definition: Let A, B, and C be any three bundles. Then #(A) denotes the number of elements in A, A 4 B := [A\B] ∪ [B\A] is the symmetric difference between A and B, #(A 4 B) is the Hausdorff distance between A and B, and [A, B, C] := (A\B) ∪ C. If B is a singleton {b}, we write [A, b, C] instead of [A, {b}, C] (and similarly if C is a singleton). It is easy to see that for any utility function u : 2Ω → R, its demand correspondence D : → Ω is upper semicontinuous when 2Ω is endowed with the Hausdorff metric). That is, if {pk } is a sequence of price vectors converging to p and A ∈ D(pk ) for all k, then A ∈ D(p). The following definition presents four closely related properties for a utility function: (GS), (SI), (NC), and (SNC). The first was originally introduced by Kelso and Crawford [5]; the other three are new. Rm +

(i)

(ii)

(iii) (iv)

Definition: A utility function u : 2Ω → R satisfies the gross substitutes condition (GS) if for any two price vectors p and q such that q ≥ p, and any A ∈ D(p), there exists B ∈ D(q) such that { a ∈ A | pa = qa } ⊂ B. has the single improvement property (SI) if for any price vector p and bundle A ∈ / D(p), there exists a bundle B such that v(A, p) < v(B, p), #(A\B) ≤ 1, and #(B\A) ≤ 1. has no complementarities (NC) if for each price vector p, and all bundles A, B ∈ D(p) and X ⊂ A, there exists a bundle Y ⊂ B such that [A, X, Y ] ∈ D(p). satisfies the strong no complementarities condition (SNC) if for all A, B ⊂ Ω and X ⊂ A, there exists Y ⊂ B such that u(A) + u(B) ≤ u([A, X, Y ]) + u([B, X, Y ]).

Remark: To check whether u has no complementarities, it is enough to consider the cases in which X ⊂ A\B. And for these cases, we only need to search among bundles Y ⊂ B\A. Suppose an agent with utility function u wants to consume a bundle A at prices p. Condition (GS) states that if the prices were increased from p to q, then the agent would 3

still want to consume the objects in A whose prices did not increase. That is, at q there is an optimal bundle B that includes all those objects (and possibly others). Condition (SI) states that any suboptimal bundle A at prices p can be strictly improved by either removing an object from it, or adding an object to it, or doing both. Suppose A and B are two optimal bundles at prices p, and an arbitrary part X is removed from A. Condition (NC) says that a new optimal bundle can be constructed with the objects that are left and a part Y of the bundle B. Finally, condition (SNC) has the following interpretation. Suppose that two identical agents have utility function u, and are endowed with bundles A and B respectively (not necessarily disjoint). Suppose agent 1 hands agent 2 a subset X of her endowment. If u has no complementarities, agent 2 should be able to return to agent 1 a subset Y of his initial endowment, so that their total utility after the swap is preserved or increased. The following piece of notation is used throughout the paper. In particular, it is used in the Appendix, where we present the proof of Theorem 1 divided into Lemmas 2 – 4, and in Section 4. Notation: If A is a bundle, eA ∈ Rm denotes its characteristic vector, whose A coordinates are eA a = 1 for a ∈ A, and ea = 0 otherwise. If A is a singleton {a}, we sometimes write ea instead of eA . Theorem 1: If u is monotone, then (GS), (SI), and (NC) are equivalent. It is easy to verify that (SNC) implies (NC), and therefore, by Theorem 1, (GS) and (SI) as well. While (SNC) is a stronger condition, it has the advantage of being stated directly in terms of the utility function rather than the demand correspondence. Kelso and Crawford [5] use (GS) to prove their main results. However, (SI) turns out to be more appropriate for our analysis (i.e., in establishing that the set of Walrasian equilibrium prices is a lattice). Lemma 5: If u is monotone and satisfies (GS), then u and v(·, p) are submodular for any price vector p. Proof: We first show that u is submodular. By Lemma 1, it is enough to show that u has decreasing marginal returns. Let α ∈ A ⊂ B ⊂ Ω. Define the price vector p as follows: pa = 0 for all a ∈ B and pa = M > u(Ω) otherwise. By monotonicity, B ∈ D(p). For each ² ≥ 0, let q(²) := p + ²eα , and define ² := max { ² | B ∈ D(q(²)) }. Since D is upper semicontinuous, B ∈ D(q(²)). By (GS), for each ² ≥ 0, there exists C ∈ D(q(²)) such that C ⊃ B\{α}. Since for each ² > ², B ∈ / D(q(²)), we must have that B\{α} ∈ D(q(²)). Again by Lemma 1, B\{α} ∈ D(q(²)). Therefore u(B) − ² = v(B, q(²)) = v(B\{α}, q(²)) = u(B\{α}).

(1)

Now, define the price vector r as follows: rα := ², ra := 0 for all a ∈ A\{α}, and ra := M for all a ∈ / A. Clearly, if X is any bundle such that X 6⊂ A, then X ∈ / D(r). Since A = { a ∈ B | ra = qa (²) }, (GS) implies that A ∈ D(r). Therefore u(A) − ² = v(A, r) ≥ v(A\{α}, r) = u(A\{α}). 4

(2)

Equations (1) and (2) imply that u(B) − u(B\{α}) = ² ≤ u(A) − u(A\{α}). Hence, u has decreasing marginal returns. Finally, v(·, p) is submodular because it is the sum of two submodular functions. Kelso and Crawford [5] provide an example showing that submodularity and monotonicity do not imply the (GS) condition. Thus, the converse of Lemma 5 is false. Definition: A utility function u represents a unit demand preference if u(∅) = 0 and for each nonempty bundle A, u(A) = max u({a}). a∈A

A unit demand utility function u is completely specified by the values it assigns to singletons and the empty set, and we will sometimes abuse notation and write u(a) instead of u({a}), for a ∈ Ω. Every unit demand utility function satisfies the (SNC) condition. Koopmans and Beckmann [6] study exchange economies where all consumers have unit demand preferences. They show that the Walrasian equilibrium problem is equivalent to the standard (linear programming) assignment problem, and that Walrasian prices coincide with its dual variables. They note that the dual problem always has a solution, and thus establish the existence of Walrasian prices. In the same setting, Leonard [8] shows that Walrasian prices can be interpreted as marginal values of the (society’s) surplus function, and discusses the incentive compatibility of a generalization of the Vickrey auction (see Section 5 below). Two other classes of functions that satisfy the (GS) condition are the set of additively separable utility functions and the set of additively concave functions. An additively separable utility function u is also completely specified by the values it assigns to singletons. Its value for any bundle A is given by u(A) =

X

u({a}).

a∈A

An additively concave utility function partitions Ω into sets of “homogeneous” goods. Suppose that there are only two distinct objects α and β, and that Ω contains several units of each: Ω = {α1 , . . . , αr , β1 , . . . , βs }. Let N denote the set of nonnegative integers. Assume that there are two increasing functions uα , uβ : N → R such that u(A) = uα (x) + uβ (y) whenever A contains x units of α and y units of β. If uα and uβ are “concave” (that is, have decreasing marginal returns), then u satisfies the (SNC) condition. Conversely, if u satisfies the (GS) condition, uα and uβ must be concave. Bevia, Quinzii and Silva [1] have introduced a class of preferences that can be represented by utility functions u satisfying the property u(A) =

X

u({a}) − c(#(A))

a∈A

5

A ⊂ Ω,

where c : N → R. Any such a utility function satisfies the (GS) condition if c is “convex” (that is, has increasing marginal returns). In addition to these classes of (GS) preferences, there are two operations that allow us to derive new (GS) preferences from other known (GS) preferences. Suppose that u1 and u2 are two (GS) functions on Ω and that there are two bundles A1 and A2 such that A1 ∩ A2 = ∅ and ui (Ai ) = ui (Ω), i = 1, 2. Then, the utility function u, defined by u(B) = u1 (B) + u2 (B) for each B ⊂ Ω, satisfies the (GS) condition. For any k < m, the k-satiation of any utility function u is the utility function u ˆ defined by u ˆ(A) = max ui (B) s.t. B ⊂ A

and

#(B) ≤ k.

If u is additively separable or additively concave, u ˆ satisfies the (GS) condition. The ksatiation of an additively separable utility function results in a natural extension of a unit demand preference. 3. Walrasian Equilibria The economy E = (Ω; u1 , . . . , un ) consists of the finite collection of objects Ω, and the set of consumers N := {1, . . . , n}. Each consumer i has a quasilinear utility function Ui : 2Ω × R → R; for each bundle A ⊂ Ω and money amount t ∈ R (she has for consumption of other goods), Ui (A, t) = ui (A) + t, where ui : 2Ω → R. Without loss of generality, we normalize so that ui (∅) = 0, and assume that each consumer i is endowed with a sufficient amount of money Mi > ui (Ω). We denote by vi consumer i’s corresponding net utility function. We also assume the economy has free disposal, and let N0 := N ∪ {0}. Note that our description of the economy does not make any reference to endowments. Due to quasilinearity and our assumption that each agent is endowed with a large amount of money, the set of Walrasian equilibrium allocations of objects and the associated prices, as well as the set of efficient allocations of objects, are independent of the initial endowments of objects. Thus, we choose to ignore the initial endowment and characterize efficiency only in terms of the allocation of objects. Walrasian equilibria are fully described by the allocation of objects, the prices of the goods and the implied transfers of money. Definition: X Sn= (X0 , . . . , Xn ) is a partition (or allocation) of Ω if (1) Xi ∩ Xj = ∅ for all i 6= j; and (2) i=0 Xi = Ω. The possibility that Xi = ∅ for some i is allowed. For i ∈ N , Xi represents consumer i’s consumption bundle, and X0 represents the collection of objects that are not consumed by anyone. Definition: The tuple (X0 , . . . , Xn ; t1 , . . . , tn ), where (Xi , ti ) represents the bundle and money amount consumed by P i, is an outcome for the economy if it satisfies the P feasibility constraints: (1) i∈N ti = i∈N Mi ; and (2) (X0 , . . . , Xn ) is a partition of Ω (since each object ω ∈ Ω can be consumed by at most one consumer). Definition: A Walrasian Equilibrium for the economy E = (Ω; u1 , . . . , un ) is a tuple (p, X), where p ∈ Rm + is a price vector, and X = (X0 , . . . , Xn ) is a partition of Ω such that (1) < p, X0 > = 0, and (2) for each i ∈ N , vi (Xi , p) ≥ vi (A, p) for all bundle A ⊂ Ω. 6

Let (p, X) be a Walrasian equilibrium. Since < p, X0 > = 0, pa = 0 for each a ∈ X0 , and if u1 is monotone, v1 (X1 ∪ X0 , p) ≥ v1 (X1 , p). That is, at prices p, X1 ∪ X0 is also an ˆ where X ˆ 0 = ∅, X ˆ 1 = X1 ∪ X0 , and X ˆ j = Xj optimal bundle for consumer 1. Thus, (p, X), for each j ≥ 2, is also a Walrasian equilibrium. Therefore, without loss of generality, we will sometimes assume that the Walrasian equilibria (p, X) we choose satisfy the additional requirement that X0 = ∅. Existence of a Walrasian equilibrium in our model is implied by Theorem 3 in Kelso and Crawford [5], that guarantees the existence of strict core allocations. We restate in our notation the definition of a strict core allocation. Definition: (p, X), where p ∈ Rm + and X is a partition of Ω, is a strict core allocation if X0 = ∅ and there does not exist an agent i, a bundle Yi , and a price vector q ≥ p, such that vi (Yi , q) > vi (Xi , p). Theorem 3 in Kelso and Crawford [5] shows that if agents’ preferences satisfy the (GS) and the (MP) condition, then a strict core allocation for E exists. Condition (MP) is equivalent in our model to the monotonicity of the agents’ utilities. Hence, if each agent’s utility satisfies (GS), E has a strict core allocation. It is easy to show that (p, X) is a strict core allocation iff (p, X) is a Walrasian equilibrium with X0 = ∅. Hence, if all preferences are monotone and satisfies the (GS) condition, then a Walrasian equilibrium exists. Theorem 2 below establishes that in a sense (GS) is a “necessary” condition for existence of Walrasian equilibrium. For any consumer with a monotone utility function u that fails the (GS) condition, one can find a collection of unit demand consumers such that the resulting economy has no Walrasian equilibrium. Thus, Theorem 2 shows that the Kelso and Crawford’s existence theorem is the strongest possible generalization of Koopmans and Beckmann’s result for unit demand economies. The proof of Theorem 2 is relegated to the Appendix. Theorem 2: Consider a consumer with a utility function u1 : 2Ω → R that violates (SI). Then, there exist ` − 1 unit demand consumers with utility functions ui , i = 2, . . . `, such that the economy E = (Ω; u1 , . . . , u` ) does not have a Walrasian equilibrium. The standard theorems of welfare economics hold for our economy E. However, for several proofs below we need the following slightly stronger second theorem of welfare economics. It is easy to see that an outcome (X, t) is Pareto efficient iff X maximizes total utility. In several proofs below we use the following lemma. Lemma 6: If p is any Walrasian price vector and Y is any efficient allocation, then (p, Y) is a Walrasian equilibrium. Proof: Suppose the allocation X is such that (p, X) is a Walrasian equilibrium. Then < p, X0 > = 0 and vi (Xi , p) ≥ vi (Yi , p) for each i ∈ N . And since Y is efficient, we have X X X ui (Yi ) − < p, Ω > ≥ ui (Xi ) − < p, Ω > = vi (Xi , p) i∈N

≥

X

i∈N

vi (Yi , p) =

X

i∈N

i∈N

ui (Yi ) − < p, Ω > + < p, Y0 >.

i∈N

7

These inequalities imply that < p, Y0 > = 0 and vi (Yi , p) = vi (Xi , p) for each i ∈ N (that is, Yi is also an optimal bundle for consumer i at prices p). Definition: Let p and q be two price vectors. Their join r = p ∨ q and meet s = p ∧ q are the price vectors defined by ra := max {pa , qa }

and sa := min {pa , qa }

for each

a ∈ Ω.

A set of price vectors P is a lattice if for V all p, q ∈ P , both W p ∨ q ∈ P and p ∧ q ∈ P . The lattice P is complete if for any Q ⊂ P , (Q) ∈ P and (Q) ∈ P , where V a

(Q) := inf {qa | q ∈ Q}

and

W a

(Q) := sup {qa | q ∈ Q}

for all a ∈ Ω.

Definition: A price vector p supports a partition X of Ω if vi (Xi , p) ≥ vi (A, p) for each bundle A and consumer i. A price vector supports a bundle A if p supports a partition X of Ω such that X0 = Ω\A. Observe that if p supports X, then (p, X) is a Walrasian equilibrium iff < p, X0 > = 0. The next result, together with Theorems 4 and 5 of Section 4, enables us to interpret Walrasian prices as shadow prices. Theorem 3: Assume ui has the (SI) property for each i ∈ N . Then, the set of prices that support a partition X of Ω is a complete lattice. Proof: Let P denote the set of all prices that support X. If P is empty, we are done. Otherwise, let p, q ∈ P and r := p ∧ q, and assume that r does not support the partition X. Then there exists i ∈ N and bundle Z such that vi (Xi , r) < vi (Z, r). By (SI), we can assume that Z = [Xi , A, B], where A ⊂ Xi , B ∩ Xi = ∅, A is either empty or a singleton {a}, and B is either empty or a singleton {b}. The inequality vi (Xi , r) < vi (Z, r) is equivalent to ui (Z) − ui (Xi ) > < r, Z > − < r, Xi > = < r, B > − < r, A >.

(∗)

If < r, A > = < p, A > ≤ < q, A > and < r, B > = < p, B > ≤ < q, B >, the above inequality implies that vi (Z, p) > vi (Xi , p), contradicting the fact that p supports X. Similarly, if < p, A > ≥ < q, A > = < r, A > and < p, B > ≥ < q, B > = < r, B >, vi (Z, q) > vi (Xi , q), another contradiction. Therefore, both A and B are nonempty and either [ra = pa < qa and rb = qb < pb ] or [ra = qa < pa and rb = pb < qb ]. Assume the former. Then < r, B > − < r, A > = qb − pa > qb − qa , and (∗) above implies that vi (Z, q) > vi (Xi , q), a contradiction. By symmetry, if we assume the latter, we obtain vi (Z, p) > vi (Xi , p), another contradiction. Therefore r = p ∧ q supports X. The proof that p ∨ q also supports X follows a similar argument, and is omitted. Hence P is a lattice. 8

For each a ∈ Ω, the projection function ϕ(p) := pa , p ∈ Rm + , is continuous. Therefore, to prove that P is complete, it is enough to show that P is closed. But p ∈ P iff it satisfies the linear constraints < p, Xi > − < p, A > ≤ ui (Xi ) − ui (A)

for all i ∈ N

and A ⊂ Ω.

That is, P is a closed simplex in Rm +. Corollary 1: Suppose ui has the (SI) property for each i ∈ N . Then, the set P W of Walrasian equilibrium prices is a complete lattice. Proof: By Kelso and Crawford [5], P W is nonempty. Suppose p, q ∈ P W and X is an efficient allocation. Then, by Lemma 6, (p, X) and (q, X) are Walrasian equilibria, and both p and q support X. By the previous Theorem, p ∨ q and p ∧ q also support X. Since < p, X0 > = < q, X0 > = 0, we have that < p ∨ q, X0 > = < p ∧ q, X0 > = 0. Therefore (p ∨ q, X) and (p ∧ q, X) are Walrasian equilibria, and P W is a lattice. Finally, for any efficient partition X, P W is equal to the set of prices p that support X and satisfy the additional linear constraint < p, X0 > = 0. Therefore, P W is a closed simplex, and thus it is a complete lattice as well. Definition: Let p := prices for E.

V

(P W ) and p :=

W

(P W ), where P W is the set of Walrasian

By Corollary 1 and the existence of Walrasian equilibria, p and p exist and are themselves Walrasian prices. With two additional conditions, (NTW) and (NTF), which are “generically” satisfied, Kelso and Crawford [5] establish that their discrete salary adjustment process converges to the best discrete core allocation for the agents (see their Theorem 4). In our context, the best discrete core allocation for the consumers is equivalent to the smallest Walrasian prices. However, the (NTW) and (NTF) conditions have no counterparts in our model since we do not consider discretized economies (i.e., a smallest unit of currency). 4. Social Surplus In this section we establish that society’s largest and smallest marginal valuations of an object coincide with the object’s largest and smallest Walrasian prices respectively. It is necessary here to study situations that involve allocations that are infeasible. That is, we need to allow for allocations X = (X1 , . . . , Xn ) ⊂ Ωn with bundles that are not necessarily pairwise disjoint. Alternatively, we can interpret such an allocation as if society’s endowment has been increased to include several identical copies of some objects. S We also need to study allocations X where Xi is a strict subset of Ω. We view such cases as if society’s endowment has been reduced to exclude some objects in Ω. Let Ω(n) denote the set that contains n identical copies of each object a ∈ Ω, and let Z := {0, 1, . . . , n}m . We endow Z with the standard partial ordering: for w, z ∈ Z, w ¹ z iff wa ≤ za for all a ∈ Ω. With this partial ordering, Z is a lattice. Each z ∈ Z represents a bundle in Ω(n) which, for each a ∈ Ω, contains za copies of a. We next extend the notion 9

of an indicator vector, defined earlier. For any bundle A ∈ Ω(n), let eA ∈ Z denote the indicator vector, whose coordinate eA a is equal to the number of copies of a contained in A, for each a ∈ Ω. As before, if A is a singleton {a}, we will write sometimes ea instead of eA . Note that for any z ∈ Z, z ∧ eΩ is a vector whose a-th coordinate is equal to 1 if za ≥ 1 and equal to 0 otherwise. We change the domain of the utility function ui from Ω to Z as follows: u∗i (eA ) := ui (A) for each A ⊂ Ω, and u∗i (z) := u∗i (z ∧ eΩ ) for any z ∈ Z. With the change of domain, we can also extend the domain of the function ui to Ω(n): for any A ∈ Ω(n), ui (A) := u∗i (eA ). The interpretation of the extension u∗i to vectors z having coordinates greater or equal to 2 is that buyer i’s utility does not increase with the consumption of additional copies of the same object, no matter what other objects she is already consuming. It is easy to verify that if ui : Ω → R satisfies the (GS) condition, then its extension ui : Ω(n) → R satisfies the (GS) condition as well. Similarly, if ui is monotone, its extension is monotone. Ω(n) endowed with the set inclusion ordering (A ¹ B iff A ⊂ B) is also a lattice. It is interesting to compare the lattices Z and Ω(n). In general, there are bundles A 6= B for which eA = eB . Thus, A ⊂ B implies eA ≤ eB , but eA ≤ eB does not imply that A ⊂ B. We now consider exchange economies E ∗ = (z; u∗1 , . . . , u∗n ) with total endowment z ∈ Z. Alternatively, we consider economies E 0 = (A; u1 , . . . , un ), with total endowment A ⊂ Ω(n). By the previous comment, if ui : Ω → R has no complementarities for each i ∈ N , then all the results of Section 3 (especially, Lemma 6 and Theorem 3) apply to the economy E 0 = (A; u1 , . . . , un ), for any A ⊂ Ω(n). Definition: The surplus function S : Z → R assigns to each m-dimensional resource vector z (with nonnegative integer coordinates) the value X X S(z) := max { u∗i (zi ) | (z1 , . . . , zn ) ∈ Zn and zi ≤ z }. i∈N

i∈N

S(z) is the total society value that can be achieved with a resource vector z ∈ Z. Prices for Ω(n) are of dimension nm, while prices for Z are of dimension m. Thus, when working with the domain Z, we are implicitly assuming that all copies of an object have the same price. However, since different copies of an object are indistinguishable for the consumers, it is intuitive that even when prices for different copies are allowed to differ, in equilibrium these must coincide. But, this result is not used until Section 5, where we formally state it and prove it in Lemma 8. Theorem 4: Suppose ui is monotone and has the (SI) property for each i ∈ N . Let p be the smallest prices that support A ⊂ Ω. Then, for each a ∈ / A, pa = S(eA +ea )−S(eA ). In particular, the smallest Walrasian prices are pa = S(eΩ + ea ) − S(eΩ ), a ∈ Ω. Proof: Pick any a ∈ / A and let qa := S(eA + ea ) − S(eA ). Consider the economy E 0 := (A ∪ {a}; u1 , . . . , un , ua ), where ua denotes the unit demand preference defined by ½ µ if b = a ua (b) = 0 otherwise, and µ is a parameter to be specified. 10

Consider first the choice µ = qa − ² for some ² > 0. Let X be an efficient allocation for E . Since µ < qa , we must have that Xa = ∅. Thus p (restricted to A ∪ {a}) together with X is a Walrasian equilibrium for E 0 . Therefore pa ≥ µ = qa − ², and since this holds for any ² > 0, we conclude that pa ≥ qa . Now consider the choice µ = qa + ² for some ² > 0. Again, let (r, X) be a Walrasian equilibrium of E 0 . By efficiency, Xa = {a}, and therefore ra ≤ µ. Let M > ui (Ω) for all i ∈ N , and define rb := M for each b ∈ / A ∪ {a}, to construct a price vector for Ω (which, with abuse of notation, we denote by the same symbol r). Then r supports A. Therefore pa ≤ ra ≤ qa + ² for all ² > 0. Hence pa ≤ qa . 0

Theorem 5: Suppose ui is monotone and has the (SI) property for each i ∈ N . Then, pa = S(eΩ ) − S(eΩ − ea ) for every a ∈ Ω. Proof: Pick any a ∈ Ω and define qa := S(eΩ ) − S(eΩ − ea ). For ua and µ as defined in the proof of Theorem 4, let E 0 := (Ω; u1 , . . . , un , ua ). We first show that pa ≤ qa by contradiction. Suppose that pa > qa and let µ := (qa + pa )/2. Let (p, X) be a Walrasian equilibrium of E, and X0 := (X, Xa ), where Xa := ∅. Since pa > µ, (p, X0 ) is a Walrasian equilibrium of E 0 . By the first theorem of welfare economics, the maximal social surplus in E 0 is equal to that of E (that is, equal to S(eΩ )). But if instead we allocate optimally Ω\{a} among the first n consumers and give a to the last consumer, then the total surplus is S(eΩ − ea ) + µ > S(eΩ − ea ) + qa = S(eΩ ), which is a contradiction. Now make µ := qa . Let X be an efficient allocation in E 0 such that Xa = ∅. By Lemma 6, p supports X. Therefore pa ≥ µ = qa . Theorems 4 and 5 generalize Leonard’s [8] results for unit demand economies. Theorem 6: Suppose each ui is monotone and has the (SI) property. Then, S : Z → R has decreasing marginal returns. Proof: It is easy to show that S is submodular iff for any z ∈ Z such that for two elements a, b ∈ Ω, za , zb < n, S(z + ea ) + S(z + eb ) ≥ S(z) + S(z + ea + eb ). Suppose z ∈ Z and a, b ∈ Ω satisfy the above conditions. Let A, B ⊂ Ω(n) be such that eA = z and eB = z + ea + eb . Define µa := S(z + ex ) − S(z) and µb := S(z + eb ) − S(z). Consider the economy E 0 := (B; u1 , . . . , un , ua , ub ), where the last two consumers have unit demand preferences and only care about objects a and b respectively. Allocate A efficiently among consumers i ∈ N , and give a to consumer a and b to consumer b; call this allocation X. If p denotes the smallest prices that supports A (in the economy E n := (Ω(n); u1 , . . . , un )), then (p, X) is a Walrasian equilibrium of E 0 . By the definition of µa , another efficient allocation in E 0 can be constructed by allocating A∪{a}1 efficiently among consumers i ∈ N , give nothing to consumer a and b to consumer b. By Lemma 6, p together with this new allocation is also a Walrasian equilibrium of E 0 . This implies that p also 1

Here A ∪ {a} denotes the bundle in Ω(n) that contains one more copy of a than A, and the same number of copies of x for any other x ∈ Ω.

11

supports A∪{a} (in E n ) and that pb ≤ µb . By Theorem 4, pb ≥ S(eA +ea +eb )−S(eA +ea ). Thus S(eA + eb ) − S(eA ) ≥ S(eA + ea + eb ) − S(eA + ea ), as was to be shown. Suppose ui : 2Ω → R is monotone and has the (SI) property for each i ∈ N . Lemma 1 then implies that S : Z → R is submodular. It can be shown, however, that if each ui is only submodular (and monotone), S may fail to be submodular. The following comparative static result is reminiscent of Topkis’ monotonicity theorem [11]. View the Walrasian equilibrium problem as parametrized by the set of objects available in the economy, and endow 2Ω (the set of parameters) with the partial order A ¹ B iff A ⊂ B. As in Topkis’ theorem, for each parameter A, the set of soA prices) lutions V P A (Walrasian V B W A associated W B with A is a complete lattice, and if A ¹ B, then P ≥ P and P ≥ P . A similar result holds if we view the Walrasian equilibrium problem as parametrized by the set of consumers. Theorem 7: Suppose each ui is monotone and has the (SI) property. For each bundle A ⊂ Ω and consumer i define the economies E A := (A; u1 , . . . , un ) and E −i := (Ω; u−i ). Let pA and pA denote respectively the smallest and largest equilibrium price vector for E A . Define p−i and p−i similarly for E −i . Then (i) if A ⊂ B ⊂ Ω, pA ≥ pB and a a A B −i −i pa ≥ pa for all a ∈ A; and (ii) p ≤ p and p ≤ p. Proof: (i) follows directly from Theorems 4, 5, and 6. Next we show that p−i ≤ p. Denote by Si the surplus function associated with E −i . Let (p−i , X−i ) be a Walrasian equilibrium for E −i , and suppose the bundle Xi maximizes vi (B, p−i ) over all B ⊂ Ω. Then, (p−i , X) is a Walrasian equilibrium for the economy E 0 = (A; u1 , . . . , un ), where A ⊂ Ω(2) is the bundle that contains two copies of each object in Xi and one copy of all other objects in Ω\Xi . By part (i), for each a ∈ Ω, = Si (eΩ + ea ) − Si (eΩ ) = Si (eΩ + ea ) + ui (Xi ) − [Si (eΩ ) + ui (Xi )] p−i a = Si (eΩ + ea ) + ui (Xi ) − S(eΩ + eXi ) ≤ S(eΩ + eXi + ea ) − S(eΩ + eXi ) ≤ pa . The second inequality in (ii) is proved analogously. With the two additional conditions (NTW) and (NTF) discussed earlier, Kelso and Crawford’s [5] Theorem 5 establishes for their discretized economy results similar to our Theorem 7. 5. Vickrey Auctions In this section we compare the outcomes of strategy-proof mechanisms studied by Vickrey [12], Clarke [3] and Groves [4] with Walrasian outcomes. In particular, we show that the Vickrey-Clarke-Groves (VCG) payment for a bundle is never greater than the value of that bundle at the smallest Walrasian prices. Moreover, we show that the gap between these two values disappears if the economy is replicated at least m + 1 times, where m is the number of different objects. Throughout this section we assume that agent 0 (the seller) initially owns all the objects and has no utility for them. 12

In discussing mechanism design issues, we need to consider the possibility that agents do not report truthfully, and thus, temporarily, we make explicit the dependence of the surplus function on the profile of utilities. Definition: consumer i, let

For a given profile of preferences u = (u1 , . . . , un ) over Ω, z ∈ Z, and S(z; u) := max {

X

uj (Xj ) |

j∈N

Si (z; u−i ) := max {

X

X

eXj ≤ z }

j∈N

uj (Xj ) |

j6=i

X

eXj ≤ z }.

j∈N

Vickrey Auction: Each buyer i submits a complete utility function ui : 2Ω → R (this is equivalent to reporting a vector of dimension 2m − 1). The seller then finds an efficient allocation X with respect to the reported profile of preferences (u1 , . . . , un ). Consumer i ∈ N receives the bundle Xi and pays the Vickrey payment qi (Xi ; u−i ), where qi (Xi ; u−i ) := Si (eΩ ; u−i ) − Si (eΩ − eXi ; u−i ). Note that the Vickrey payments depend on the efficient allocation chosen, and that there might be several efficient allocations associated with the same utility profile u = (u1 , . . . , un ). It is well known that the buyers and the seller are indifferent about which efficient allocation is chosen when every buyer reports his true preferences (see, for example, Krishna and Perry [7]), and that the VCG mechanism is strategy-proof. For the rest of this section, we assume that the agents report truthfully their preferences, and drop the profile u from the list of arguments in the functions S, Si , and qi , i ∈ N . Consider the case where all consumers have unit demand preferences. We only need to consider allocations that assign at most one object to each consumer. Thus, the Vickrey payments are defined for objects. Leonard [8] has shown that in this case, the Vickrey payments coincide with the smallest Walrasian prices. It is easy to extend this result to the case in which each consumer i has “linear preferences” of the form X ui (A) = ui ({a}), A ⊂ Ω. a∈A

However, the following example shows that with more general utility functions that have the (SI) property, this result typically does not hold. There are three identical objects and two consumers with the same preferences. For i = 1, 2, ui (A) is equal to 0 if #(A) = 0, to 10 if #(A) = 1, to 18 if #(A) = 2, and to 20 if #(A) = 3. Since the objects are indistinguishable, in any equilibrium their prices must coincide. All efficient allocations assign one object to one consumer and two objects to the other. Therefore, (8, 8, 8) is the unique Walrasian price vector. The Vickrey payment for the consumer getting one object is 20 − 18 = 2 < 8, and for the consumer getting two objects is 20 − 10 = 10 < 8 + 8 = 16. Thus, each consumer is paying strictly less in the Vickrey auction than in (any) Walrasian equilibrium. Although the equality is not attained in general, the next theorem establishes that even without the (GS) condition, a consumer’s Vickrey payment for her bundle is never more than the value of that bundle at the smallest Walrasian prices. 13

Theorem 8: Let (p, X) be a Walrasian equilibrium of E = (Ω; u1 , . . . , un ). Suppose each ui is monotone. Then, < p, Xi > ≥ qi (Xi ) for each i ∈ N . Proof: Consider the economy E 0 = (Ω; u1 , . . . , ui−1 , u0i , ui+1 , . . . , un ), where consumer i is replaced by a consumer with linear preferences, given by X u0i (A) = pa . a∈A∩Xi

It is easy to see that (p, X) is also a Walrasian equilibrium of E 0 , with associated total surplus S 0 = < p, Xi > + Si (eΩ − eXi ). Now consider the economy E 00 , where consumer i is replaced by a consumer with utility function u00i (A) = 0 for all A. Its total surplus is S 00 = Si (eΩ ). Obviously S 0 ≥ S 00 , and by definition, Si (eΩ ) = qi (Xi ) + Si (eΩ − eXi ). Hence, < p, Xi > ≥ qi (Xi ). Makowski and Ostroy [9] prove that in a quasilinear economy the private marginal product of each agent is no greater than his social marginal product. Straightforward manipulations of their definitions of private and social marginal product reveal this result to be equivalent to Theorem 8 above. The next results deal with replica economies. For any k ∈ N, the k-replica of ˆ with set of objects Ω(k) containing k economy E = (Ω; u1 , . . . , un ) is the economy E identical copies of each object in Ω, and k “copies” i1, . . . , ik of each consumer i. The utility function of a consumer ij is defined as follows. For any bundle A ⊂ Ω(k), let uij (A) := u∗i (eA ∧ eΩ ) (as defined in the previous section).2 ˆ in which each If X is an allocation for E, then Xk denotes the allocation for E ˆ then pˆ = consumer type i receives the same bundle Xi . If pˆ is a price vector in E, 1 k j (ˆ p , . . . , pˆ ), where pˆ represents the prices for the j-th copy of each object. Lemma 7: Suppose ui is monotone and has no complementarities for each i ∈ N . If X is an efficient allocation for E, then Xk is an efficient allocation for its k-replica economy ˆ and if p is a Walrasian price vector for E, then pˆ = (p, . . . , p) is a Walrasian price vector E, ˆ Conversely, if pˆ is a Walrasian price vector for E, ˆ there exists a price vector p in E for E. such that pˆ = (p, . . . , p). Proof: Let X be an efficient allocation for E. By Lemma 6, if p is any Walrasian price vector for E, then (p, X) is a Walrasian equilibrium. It is easy to see that ˆ Therefore, by the first theorem of welfare ((p, . . . , p), Xk ) is a Walrasian equilibrium for E. ˆ economics, Xk is an efficient allocation for E. ˆ is a Walrasian equilibrium for E. ˆ Then, by the definition of Now, suppose (ˆ p, X) ˆ ˆ ij contains the preferences of the consumers in E, we can assume wlog that for each ij, X 0 at most one copy of a, for every a ∈ Ω. Therefore, for any two copies a and a00 of the same object a, pˆa0 = pˆa00 , for otherwise the individual consuming the most expensive copy would rather consume the cheapest copy instead. 2

Although in Section 4 we only required the case k = n, it is clear that the preferences’ extensions and the notation defined there apply to any k ≥ 2.

14

ˆ by Lemma 7 and with abuse Since we will only consider Walrasian prices pˆ for E, of notation we will view pˆ as an m-dimensional vector only. Corollary 2: Pˆ W = P W . We have proven above that the smallest Walrasian prices p are always an upper bound for the set of Vickrey payments, in the sense that the value of consumer i’s bundle at prices p is never less than its corresponding Vickrey payment. The next theorem shows that if the economy is replicated at least k = m + 1 times, then the Walrasian prices ˆ we denote by Sˆ and Sˆij “coincide” with the Vickrey payments. For each consumer ij of E, ˆ and for any bundle A ⊂ Ω, let qˆij (A) := Sˆij (keΩ )− Sˆij (keΩ −eA ) the surplus functions of E, (note that eΩ(k) = keΩ ). As before, we omit the profile of preferences because it is assumed to be fixed at the true profile. Theorem 9: Suppose that ui is monotone and has no complementarities for each ˆ is any efficient allocation of ˆ be the k-replica economy. If X i ∈ N . Let k ≥ m + 1 and E k ˆ ij ) = ˆ E (not necessarily of the form X for some efficient allocation X of E), then qˆij (X ˆ ij > for each replica ij of consumer i, and each i ∈ N . < p, X ˆ without Proof: Let Y be any efficient allocation in E, and pick any consumer in E; loss of generality, and to simplify the notation, assume this is a consumer rk in the last cohort. By Lemma 7, we have ˆ Ω(k) ) = Sˆrk (eΩ(k) − eXˆ rk ) + urk (X ˆ rk ). k · S(eΩ ) = S(e (1) One possible allocation of Ω(k) among the consumers excluding rk is obtained as follows. Suppose Yr = {a1 , . . . , al }. For each j = 1, . . . , l, distribute Ω ∪ {aj } efficiently among the consumers in the j-th cohort. For each j = l + 1, . . . , k − 1, give each consumer ij, i ∈ N , the bundle Yi . Finally, for the last cohort, give each consumer ik, excluding consumer rk, the bundle Yi . This allocation has total surplus l X

S(eΩ + eaj ) + (k − l − 1)S(eΩ ) + [S(eΩ ) − ur (Yr )] = kS(eΩ ) + < p, Yr > − ur (Yr ).

j=1

Therefore, and

Sˆrk (Ω(k)) ≥ kS(eΩ ) + < p, Yr > − ur (Yr ),

(2)

ˆ rk ) = Sˆrk (eΩ(k) ) − Sˆrk (eΩ(k) − eXˆ rk ) qˆrk (X ˆ rk )] ≥ [kS(eΩ ) + < p, Yr > − ur (Yr )] − [kS(eΩ ) − urk (X ˆ rk ) − ur (Yr ) + < p, Yr >, = urk (X

ˆ is efficient, wlog we can assume where the last inequality follows from (1) and (2). Since X ˆ ˆ rk ⊂ Ω, and therefore urk (X ˆ rk ) = ur (X ˆ rk ). Again by Lemma 7, (p, Y) and (p, X) that X ˆ Thus are respectively Walrasian equilibria of E and E. ˆ rk ) − < p, X ˆ rk >. ur (Yr ) − < p, Yr > = ur (X ˆ rk ) ≥ < p, X ˆ rk >. TheSubstituting this equality in the previous inequality, we get qˆrk (X ˆ rk ) = < p, X ˆ rk >. orem 8 then implies that qˆrk (X 15

6. Production In this section we introduce a production technology that satisfies a condition, No Complementarities in Production (NCP), analogous to the (NC) condition. We show how a production economy endowed with this technology can be identified with an exchange economy satisfying the (GS) condition. We then use this construction to extend results from preceding sections to economies with production. Suppose that there are ` firms in the economy, and define L := {1, . . . , `}. Let Ω denote the maximal collection of objects (including multiple units or copies) that the agents would ever consume collectively. Without any assumptions, the set Ω may be infinite. We will assume below that production costs are “convex”, and that there exists a set Ω sufficiently large so that the marginal surplus for the consumers (when they consume Ω) of any additional (unit of an) object is less than the marginal cost of producing that object (when the firms are already producing Ω efficiently). As before, m := #(Ω). Firm k is totally characterized by its cost function ck : 2Ω → R+ . We will require that each ck be monotone and have no production complementarities (as defined below). Definition: Firm k’s profit function πk : 2Ω × Rm + → R and supply correspondence m Ω Σk : R+ → 2 are defined by πk (A, p) := < p, A > − ck (A) A ⊂ Ω, p ∈ Rm +, Σk (p) := { A | πk (A, p) ≥ πk (B, p) for all B ⊂ Ω }

p ∈ Rm +.

Definition: The cost function ck : 2Ω → R+ is monotone if ck (A) ≥ ck (B) for all A ⊃ B, and has no production complementarities (NPC) if for every A, B ∈ Σk (p) and X ⊂ A\B, there exists Y ⊂ B\A such that [A, X, Y ] ∈ Σk (p). Definition: (p; X1 , . . . , Xn ; Y1 , . . . , Y` ) is a Walrasian equilibrium for the production economy E P = (Ω; u1 , . . . , un ; c1 , . . . , c` ) if (1) vi (Xi , p) ≥ vi (A, p) for all A ⊂ Ω and i ∈ N . (2) πk (Yk , p) ≥ πk (A, p) for all A ⊂ Ω and k ∈ L. P P Xi Yk e ≤ (the difference of these two vectors represents the set of (3) i∈N k∈L e objects that are produced and not consumed). A firm that produces Ac = Ω\A can be viewed as “consuming” the bundle A. To construct a Walrasian equilibrium, we will transform every firm into a consumer and expand the set of objects in the economy to Ω(`). P Definition: Firm k’s utility function uP k , net utility function vk , and demand correspondence DkP are defined as follows: c uP k (A) := ck (Ω) − ck (A ) A ⊂ Ω, c vkP (A, p) := uP k (A) − < p, A > = ck (Ω) − ck (A ) − < p, A >

DkP (p) := { A | vkP (A, p) ≥ vkP (B, p) for all B ⊂ Ω }

A ⊂ Ω, p ∈ Rm +,

p ∈ Rm +.

Clearly, πk (Ac , p) − πk (Ω, p) = vkP (A, p). Thus, A maximizes firm k’s net utility at prices p iff Ac maximizes firm k’s profits at prices p. That is, A ∈ DkP (p) iff Ac ∈ Σk (p). 16

Lemma 8: If ck has no production complementarities then uP k has no complemenP tarities, and if ck is monotone then uk is monotone. Proof: Suppose ck is monotone and A ⊃ B. Then Ac ⊂ B c and uP k (A) = ck (Ω) − P ck (A ) ≥ ck (Ω) − ck (B c ) = uP (B). Hence u is monotone. k k Now, suppose A, B ∈ DkP (p) and X ⊂ A\B. Then Ac , B c ∈ Σk (p) and X ⊂ B c \Ac . ˆ ⊂ B c \Ac . By (NPC), there exists Yˆ ⊂ Ac \B c = B\A ˆ := A ∩ B c ∩ X c ; clearly X Let X ˆ Yˆ ] ∈ Σk (p). But, such that [B c , X, c

ˆ Yˆ ]c = [(B c ∩ X ˆ c ) ∪ Yˆ ]c = (B ∪ X) ˆ ∩ Yˆ c [B c , X, = [B ∪ (A ∩ B c ∩ X c )] ∩ Yˆ c = [(A ∪ B)\X] ∩ Yˆ c = [A, X, (B\Yˆ )] ˆ Yˆ ]c ∈ DP (p). Let Y := B\Yˆ ⊂ B; then [A, X, Y ] = [B c , X, k S ∗ View Ω(`) as Ωk , where each Ω∗k is a different copy of Ω. Ω∗k represents the production set of firm k. Let Π : Ω(`) → Ω be the projection map that to any k and “copy” a∗k ∈ Ω∗k of a ∈ Ω, assigns its “original” object a. Then, for any A ⊂ Ω(`) and k, let A∗k := A ∩ Ω∗k and Ak := Π(A∗k ) ⊂ Ω. We extend the consumers’ utilities as in Section 4. For each A ⊂ Ω(`) and i ∈ N , ui (A) := ui (Π(A)). The producers’ utilities P on Ω(`), however, are defined in a different way: uP k (A) := uk (Ak ) for any A ⊂ Ω(`) and P k ∈ L. If the original uk is monotone and/or has no complementarities, the new uP k just defined has the same properties. We are abusing notation here, since we denote by the same symbol the utilities on Ω and on Ω(`). Notice that the producers utilities on Ω(`) are not “extensions” of their utilities on Ω: producer k has positive utility only for copies in Ω∗k . To study the existence and properties of Walrasian equilibria of the production P economy, we consider the exchange economy E = (Ω(`); u1 , . . . , un , uP 1 , . . . , u` ). In this exchange economy we refer to agent i (i = 1, . . . , n) as consumer i, and to agent n + k (k = 1, . . . , `) as producer k. Theorem 10: Assume each ui and each ck in the economy E P is monotone and has no (production) complementarities. Then E P has a Walrasian equilibrium. Moreover, the set of Walrasian equilibrium prices for E P is a complete lattice. Proof: By Kelso and Crawford [5], the exchange economy E has a Walrasian equilibrium ((p1 , . . . , p` ); X0∗ , . . . , Xn∗ , X1P , . . . , X`P ). Since each producer k assigns no value to objects outside Ω∗k , we can assume without loss of generality that XkP ⊂ Ω∗k for each p k. Let Yk := Π(Ω∗k \XkP ), k ∈ L. Then, eYk = eΩ − eXk (recall that for any B ⊂ Ω(`) ∗ and a ∈ Ω, eB a is the number of copies of a contained in B). Similarly, let Xi := Π(Xi ), i ∈ N . By the way the consumers’ preferences are extended to Ω(`), we can assume that ∗ eXi = eXi for each i ∈ N (that is, each consumer does not get more than a copy of each object). Now, since (X0∗ , . . . , Xn∗ , X1P , . . . , X`P ) is a partition of Ω(`), `eΩ =

X i∈N0

∗

eXi +

X

∗

P

eXk = eX0 +

X i∈N

k∈L

17

eXi +

X

[eΩ − eYk ].

k∈L

That is,

X k∈L

∗

eYk = eX0 +

X

eXi ≥

i∈N

X

eXi .

i∈N

The price vector pk denotes the prices of objects in Ω∗k (charged by producer k). Since consumers i ∈ N find the objects in Ω∗k equivalent to the objects in Ω∗j for any j 6= k, we must have pk = pj . Suppose to the contrary that for some j 6= k and object a, pka > pja ≥ 0. Then object a∗k is consumed by some consumer i (because its price is positive). But i is indifferent between a∗j and a∗k , and a∗j is cheaper; this is a contradiction. Therefore pk = p1 for all k > 1. Since Xi∗ is optimal for i in Ω(`) at prices (p1 , . . . , p1 ), Xi is optimal for i in Ω at prices p1 . Similarly, XkP is optimal for k in Ω(`) at prices (p1 , . . . , p1 ) iff Yk is profit maximizing in Ω for firm k at prices p1 . Therefore, (p1 ; X1 , . . . , Xn ; Y1 , . . . , Y` ) is a Walrasian equilibrium of the production economy E P . It is straightforward to extend to production economies Theorem 3 and Corollary 1. 7. Conclusion In this paper we have studied the problem of efficient production and allocation when the commodity space consists of m indivisible goods and one divisible good (money). The key assumptions are the quasilinearity in the divisible good, the (GS) condition, and that each consumer is endowed with a sufficient amount money. Within this setting, we were able to provide an analysis of Walrasian equilibrium. We also established a relationship between Walrasian equilibrium and strategy-proof mechanisms. Two of the three main assumptions of the model that have been developed in this paper are familiar from auction theory. Quasilinearity in money and the fact that agents are endowed with a significant amount of money are standard assumptions in the literature. In a companion paper we build on this connection to auction theory, and study a dynamic auction/tˆ atonnement process when preferences and cost functions satisfy the (GS) condition. 8. Appendix The next three lemmas show that if u is monotone, then the three conditions (GS), (NC), and (SI) are equivalent. Lemma 2: If u is monotone, (GS) implies (SI). Proof: Pick a price vector p, and let A ∈ / D(p). For any price vector q define H(q) := { B ⊂ Ω | v(B, q) > v(A, q) } H1 (q) := { B ∈ H(q) | #(B\A) ≤ #(C\A) for all C ∈ H(q) } H2 (q) := { B ∈ H(q) | #(A\B) ≤ #(A\C) for all C ∈ H(q) }. 18

H(q) are the bundles that have strictly higher net utility than A at prices q. Since A ∈ / D(p), D(p) ⊂ H(p), and therefore H1 (p) is nonempty. We first show that for any B ∈ H1 (p), #(B\A) ≤ 1. Let B ∈ H1 (p) and pˆ be the price vector such that pˆa = pa for all a ∈ A ∪ B and pˆa = M > u(Ω) otherwise. Observe that ∅ 6= D(ˆ p) ⊂ H(ˆ p) ⊂ H(p). Pick any C ∈ H(ˆ p). Then C ⊂ A ∪ B, and therefore C\A ⊂ B\A. That is, C ∈ H1 (p). Hence H(ˆ p) ≡ H1 (ˆ p) ⊂ H1 (p). To conclude, it is enough to show that #(C\A) ≤ 1. By contradiction, suppose #(C\A) ≥ 2. Pick {x, y} ⊂ C\A, and for each ² ≥ 0, let q(²) := pˆ + ²e{x,y} . Let ∆ := { ² ≥ 0 | A ∈ / D(q(²)) and C ∈ D(q(²)) }. Since q(0) = pˆ, 0 ∈ ∆. Let ² := sup ∆. Since D is upper semicontinuous, C ∈ D(q(²)). / D(q(²)) or A ∈ D(q(²)). Assume first There are two possibilities at ²: either A ∈ the former. Then, there exists ²ˆ > ² such that A, C ∈ / D(q(ˆ ²)). Pick any X ∈ D(q(ˆ ²)); then X ⊂ A ∪ B, and either x ∈ / X or y ∈ / X (or both). Moreover, v(X, p) ≥ v(X, q(ˆ ²)) > v(A, q(ˆ ²)) = v(A, p). Therefore, X ∈ H(p) and #(X\A) ≤ #(C\A) − 1, which contradicts the fact that C ∈ H1 (p). Alternatively, now assume that A ∈ D(q(²)). Let r(²) := / D(r(²)). By (GS), there exists q(²) + ²ex . Note that for all ² > 0, A ∈ D(r(²)) and C ∈ X such that C\{x} ⊂ X ∈ D(r(²)) for all ² ≥ 0. Moreover, since C ∈ / D(r(²)) for all ² > 0, x ∈ / X. Pick any ² > 0; then v(A, p) = v(A, r(²)) = v(X, r(²)), and since y ∈ X, v(X, r(²)) < v(X, p). Consequently, X ∈ H(p) and #(X\A) ≤ #(C\A) − 1, which is again a contradiction. We have thus shown that #(B\A) ≤ 1 for all B ∈ H1 (p). For the rest of the proof, fix B ∈ H1 (p) and define pˆ as follows: pˆa = pa for all a ∈ A ∪ B and pˆa = M > u(Ω) otherwise. Pick E ∈ H2 (ˆ p) and define p0 as follows: p0a = 0 for a ∈ A ∩ E, and p0a = pˆa otherwise. Recall that H(ˆ p) ≡ H1 (ˆ p) ⊂ H1 (p). Therefore, H2 (ˆ p) ⊂ H1 (ˆ p), and thus E ∈ H1 (ˆ p) as well. Hence, E\A = B\A. To finish the proof, we show that #(A\E) ≤ 1. p) More specifically, assume that #(A\E) > 1; we then show that there exists G ∈ H1 (ˆ such that #(A\G) < #(A\E), which is a contradiction. Observe that if X ∈ H(p0 ), then 0 < v(X, p0 ) − v(A, p0 ) ≤ v(X, pˆ) − v(A, pˆ). So X ∈ H(ˆ p) ≡ H1 (ˆ p), and therefore H(p0 ) ≡ H1 (p0 ) ⊂ H1 (ˆ p). Thus, X\A = E\A = B\A. 0 Also, E ∈ H2 (p ) and A ∈ / D(p0 ). We now show that E ∈ D(p0 ). By contradiction, suppose that v(X, p0 ) > v(E, p0 ); ∗ let X := X ∪ (A ∩ E). Then, v(X ∗ , p0 ) > v(E, p0 ). Hence, X ∗ 6= E, X ∗ ⊂ A ∪ E, and A ∩ E ⊂ X ∗ . Thus, #(A\X ∗ ) < #(A\E), contradicting the fact that E ∈ H2 (p0 ). Therefore, E ∈ D(p0 ). For ² ≥ 0, let q(²) = p0 − ²eA\E . Define ∆ := { ² ≥ 0 | q(²) ≥ 0, A ∈ / D(q(²), and E ∈ D(q(²)) }. Since E ∈ H2 (p0 ), p0a > 0 for all a ∈ A\E. Also, since A ∈ / D(p0 ), A ∈ / D(q(²)) for all ² > 0 sufficiently small. Let ² := sup ∆. We show that ² > 0. If not, E ∈ / D(q(²)) for all ² > 0, and therefore there exists F such that v(F, q(²)) > v(E, q(²)) for all ² > 0 sufficiently small. Let F ∗ := F ∪ (A ∩ E). Then, v(F ∗ , q(²)) > v(E, q(²)) for all small ² > 0. By continuity, since E ∈ D(p0 ) and q(0) = p0 , F ∗ ∈ D(p0 ) ⊂ H(p0 ). Also F ∗ 6= E, and by 19

earlier remark, F ∗ \A = E\A. Therefore, #(A\F ∗ ) < #(A\E), which contradicts the fact that E ∈ H2 (p0 ). Hence, ² > 0. At ² one of three things happen: (i) qx (²) = 0 for some x ∈ A\E; (ii) A ∈ / D(q(²)) and E ∈ / D(q(²)) for all ² > ²; (iii) A ∈ D(q(²)). In case (i), make G := E ∪ {x}. In case (ii), there exists G ∈ D(q(²)) such that A ∩ E ⊂ G ⊂ A ∪ B and G 6= E. In these two cases v(G, q(²)) = v(E, q(²)) > v(A, q(²)) and #(A\G) < #(A\E). But, v(G, q(²)) − v(A, q(²)) = v(G, p0 ) − v(A, p0 ), so G ∈ H(p0 ), which contradicts the fact that E ∈ H2 (p0 ). Finally, in case (iii), A ∈ D(q(²)). Since by assumption #(A\E) > 1, there exist x 6= y such that {x, y} ⊂ A\E. Define the price vector r by: rx = p0x and ra = qa (²) for a 6= x. By (GS), there exists G ∈ D(r) such that A\{x} ⊂ G ⊂ A ∪ E. Now, A ∈ / D(r) because otherwise A ∈ D(q(²)) for some ² < ², contradicting the definition of ². Hence, v(G, r) > v(A, r). But, v(G, r) − v(A, r) = v(G, p0 ) − v(A, p0 ), so G ∈ H(p0 ), contradicting the fact that E ∈ H2 (p0 ). Lemma 3: If u is monotone, then (SI) implies (NC). Proof: Fix a price vector p, and let A, B ∈ D(p) and X ⊂ A\B. Define F := { F ∈ D(p) | F ⊂ A ∪ B and A\X ⊂ F }. Note that A ∈ F, so F = 6 ∅. Let E ∈ argmin { #(F ∩ X) | F ∈ F }. If E ∩ X = ∅, we are done: define Y := E ∩ B and note that E = [A, X, Y ]. Otherwise, suppose E ∩ X 6= ∅; we show that this leads to a contradiction. For each ² ≥ 0, define the price vector q(²) as follows: qa (²) = M > u(Ω) for a ∈ / A∪B, qa (²) = pa for a ∈ (A∪B)\X, and qa (²) = pa +² for a ∈ X. Observe that v(F, q(²)) = v(F, p)−#(F ∩X)·² for all F ⊂ A ∪ B. Thus, for all ² > 0, B ∈ D(q(²)) and v(B, q(²)) > v(E, q(²)). Hence, there exists F ⊂ A∪B such that #(E\F ) ≤ 1 and #(F \E) ≤ 1, and v(F, q(²)) > v(E, q(²)) for all ² > 0 sufficiently small. Since D is upper semicontinuous, F ∈ D(p). Now, v(F, p) − #(F ∩ X) · ² = v(F, q(²)) > v(E, q(²)) = v(E, p) − #(E ∩ X) · ², and v(F, p) = v(E, p) imply that #(F ∩ X) < #(E ∩ X). Since #(E\F ) ≤ 1, E\X ⊂ F and A\X ⊂ F . Thus, F ∈ F and #(F ∩ X) < #(E ∩ X), which contradicts the definition of E. Lemma 4: If u is monotone, then (NC) implies (GS). Proof: Let p and q be two price vectors such that q ≥ p, and define C := { a ∈ Ω | qa > pa }. The proof is by induction on the cardinality of C. Suppose #(C) = 1. Hence, C = {α} for some α ∈ Ω, and q = p + (qα − pα )eα . Pick any A ∈ D(p), and define ² := sup {² | A ∈ D(p + ²eα ) }. Since D is upper semicontinuous, A ∈ D(p+²eα ) for all ² ∈ [0, ²]. Thus, if ² ≥ qa −pa , then A ∈ D(q), and we can choose B = A. (In particular, note that if α ∈ / A, then ² = +∞.) 20

Suppose now that ² < qa − pa (so α ∈ A). Note that if ² > ² and E ∈ D(p + ²eα ), then α ∈ / E, for otherwise 0 ≤ v(A, p) − v(E, p) = v(A, p + ²eα ) − v(E, p + ²eα ) ≤ 0, which contradicts the definition of ². Since Ω is finite, there exists E ⊂ Ω and a monotone sequence {²k } of positive numbers converging to 0 such that E ∈ D(p+(²+²k )eα ) for all k. Again, the upper semicontinuity of D implies that E ∈ D(p + ²eα ). Since A ∈ D(p + ²eα ) as well, there exists Y ⊂ E such that [A, C, Y ] ∈ D(p + ²eα ). Since α ∈ / [A, C, Y ] =: B, α B ∈ D(p + ²e ) for all ² ≥ ². In particular, B ∈ D(q), and clearly B ⊃ A\C, as desired. Suppose now that the result holds whenever #(C) ≤ k, and assume that #(C) = k + 1. Pick any α ∈ C and define the price vector q˜ as follows: q˜α = pα and q˜a = qa ˜ = k, by inductive for all a 6= α. Let C˜ := C\{α}, and pick any A ∈ D(p). Since #(C) ˜ ˜ ˜ hypothesis, there exists B ∈ D(˜ q ) such that B ⊃ A\C. By inductive hypothesis again, ˜ there exists B ∈ D(q) such that B ⊃ B\{α} ⊃ A\C. Theorem 2: Consider a consumer with a utility function u1 : Ω → R that violates SI. Then, there exist ` − 1 unit demand consumers with utility functions ui , i = 2, . . . `, such that the economy E = (Ω; u1 , . . . , u` ) does not have a Walrasian equilibrium. Proof: By assumption, there exist a price vector p and a set A ∈ / D1 (p) such that for all C ⊂ Ω with #(A\C) ≤ 1 and #(C\A) ≤ 1 we have that v1 (A, p) ≥ v1 (C, p). That is, A is not optimal (at prices p), but no single switch can improve A. Consider the optimization problem argmin #(A 4 C) s.t. v1 (C, p) > v1 (A, p). Since A ∈ / D1 (p), its feasible set is nonempty; let B be an optimal solution. Then, by assumption, either (i) #B\A > 1 or (ii) #A\B > 1. Assume (i). Let k = #B\A and ² = [v1 (B, p) − v1 (A, p)]/2k > 0. We now introduce a collection of unit demand consumers. There is a special consumer, indexed by 2, and one consumer for each a ∈ / A ∩ B. Thus, N = {1, 2} ∪ [Ω\(A ∩ B)] will be the set of consumers in the economy E. Their utility functions are defined as follows. For each a ∈ Ω\(A ∩ B), let ½ ua (C) =

  pa sa = pa + ²  u1 (Ω) + 1

where

if a ∈ C otherwise,

sa 0

if a ∈ A\B if a ∈ B\A if a ∈ Ω\(A ∪ B).

For consumer 2, define ra = pa + u1 (Ω) + 1 for each a ∈ B\A, and n u2 (C) =

max {ra | a ∈ C ∩ (B\A)} if C ∩ (B\A) 6= ∅ 0 otherwise. 21

Assume that a Walrasian equilibrium (t, Y ) exists for the economy E. Since consumer 2 only values objects in B\A, we can assume wlog that Y2 ⊂ B\A. Define q as follows: qa = ta for a ∈ / A, qa = 0 for a ∈ A ∩ B, and qa = pa for a ∈ A\B. For each / Ya ), and if ta a ∈ A\B, either a ∈ Y1 or a ∈ Ya . If a ∈ Y1 , then ta ≥ pa (since a ∈ were decreased to pa , Y1 and Ya would remain optimal for consumers 1 and a respectively. Similarly, if a ∈ Ya , then ta ≤ pa , and if ta were increased to pa , Y1 and Ya would remain optimal for consumers 1 and a respectively. Moreover, all consumers, except (possibly) consumer 1, give zero value to objects in A ∩ B. Therefore, (q, Y ) is also a Walrasian equilibrium of E. And, if we define the allocation X such that X1 = Y1 ∪ (A ∩ B) and Xi = Yi \(A ∩ B) for i 6= 1, (q, X) is also a Walrasian equilibrium of E. Clearly each agent a ∈ Ω\(A ∪ B) must consume a in equilibrium, that is a ∈ Xa for all a ∈ Ω\(A ∪ B). Therefore A ∩ B ⊂ X1 ⊂ A ∪ B. Agent 2 must be consuming some a ∈ B\A, otherwise qa ≥ pa + u1 (Ω) + 1 for all a ∈ B\A. But at those prices, nobody else wants to consume any a ∈ B\A. Therefore, B\X1 6= ∅. Suppose (B\A) ∩ X1 6= ∅. Since A ∩ B ⊂ X1 , we have that #(A 4 X1 ) < #(A 4 B), and by the minimality of B, it follows that v1 (X1 , p) ≤ v1 (A, p). For each a ∈ (B\A) ∩ X1 , a∈ / Xa and therefore qa ≥ pa + ². Hence, v1 (X1 , q) < v1 (A, q), which is a contradiction. Therefore, X1 ⊂ A. We finally show that X1 ⊂ A also leads to a contradiction. Assume that X1 ⊂ A. Then each a ∈ B\A is either consumed by agent 2 or agent a, and agent 2 consumes at most one object. We first show that qa ≤ pa + ² for all a ∈ B\A. Note that if a ∈ B\A is consumed by agent a, then qa ≤ pa + ². Hence, if X2 = ∅, qa ≤ pa + ² for all a ∈ B\A. If X2 = {b}, since k ≥ 2 by assumption, there exists a ∈ B\A such that a 6= b. The optimality of X2 implies that pb + u1 (Ω) + 1 − qb = rb − qb ≥ ra − qa = pa + u1 (Ω) + 1 − qa , and since qa ≤ pa + ², we must have that qb ≤ pb + ², as desired. Now, X1 ⊂ A and A∩B ⊂ X1 imply that #(A4X1 ) < #(A4B), and by the minimality of B, we must have that v1 (X1 , p) ≤ v1 (A, p). But qa = 0 for a ∈ A ∩ B and qa = pa for a ∈ A\B. Therefore, v1 (X1 , q) ≤ v1 (A, q). Also qa ≤ pa + ² for all a ∈ B\A implies that v1 (A, q) < v1 (B, q). Thus, v1 (X1 , q) < v1 (B, q), a contradiction. We have shown that (i) contradicts the existence of equilibrium. Next, assume (ii) #(A\B) > 1 and #(B\A) ≤ 1. Now we let the set of consumers be N = {1, 2} ∪ [Ω\(A ∪ B)] if B\A = ∅ and N = {1, 2, 3}∪[Ω\(A∪B)] if B\A is a singleton. Note that in the former case, A∩B∩N = ∅, while in case (i) before we defined N so that A∩B ⊂ N . The utility functions of consumers a ∈ [Ω\(A ∪ B)] are defined as before. Consumer 2 now has utility function ½ u2 (C) =

0 max {pa + u1 (Ω) + 1 | a ∈ C ∩ (A\B)} 22

if C ∩ (A\B) = ∅ otherwise.

When B\A is a singleton {b}, we define ½ u3 (C) =

0 if C ∩ (A 4 B) = ∅ max {pa + u1 (Ω) + 1 | a ∈ C ∩ (A 4 B)} otherwise.

Again by contradiction, assume that (q, X) is a Walrasian equilibrium for the economy with consumers N . As argued above, we can assume wlog that A ∩ B ⊂ X1 ⊂ A ∪ B, and qa = 0 for all a ∈ A ∩ B. Finally, if #(X2 ) > 0, then the marginal utility of at most one object in X2 is strictly positive for player 2. Hence all remaining objects must have 0 price and can be given to any player without upsetting the equilibrium. When agent 3 exits, the same argument can be applied to him. So, we will assume wlog that #(X2 ) ≤ 1 and #(X3 ) ≤ 1. We now show that #(X2 ) = 1, and if agent 3 exists, then X3 = {b} as well. To see this note that #(X2 ) = 0 implies qa > u1 (Ω) for all a ∈ A\B, so agent 1 is not consuming any a ∈ A\B either, which is a contradiction. If 3 exists and does not consume b, then a similar argument yields a contradiction. Since A ∩ B ⊂ X1 and b ∈ / X1 , we have that v1 (X1 , p) ≤ v1 (A, p) < v1 (B, p). By assumption, #(A\B) > 1, so #(X2 ) = 1 implies that X1 ∩ (A\B) 6= ∅. Let ² = min {qa − pa | a ∈ X1 ∩ (A\B)}, and c be any optimal solution of this problem. If a ∈ (A ∪ B)\X1 , that is, if a is consumed by player 2 (or 3), then pa + u1 (Ω) + 1 − qa ≥ pc + u1 (Ω) + 1 − qc . Hence, qa ≤ pa − pc + qc ≤ pa + ². So qa < pa for some a ∈ X1 ∩ (A\B) (i.e., ² < 0) implies qa < pa for all a ∈ (A ∪ B)\X1 . But then v1 (X1 , p) ≤ v1 (A, p) implies v1 (X1 , q) < v1 (A, q), a contradition. Therefore, qa ≥ pa for all a ∈ X1 ∩ (A\B) and ² ≥ 0. If B\A = {b}, the optimality of X3 = {b} implies that pb + u1 (Ω) + 1 − qb ≥ pc + u1 (Ω) + 1 − qc . Thus, qb ≤ pb +², and v1 (B, q) ≥ v1 (B, p)−² whether B\A is the empty set or the singleton {b}. Since X1 ∩ (A\B) 6= ∅, v1 (X1 , q) ≤ v1 (X1 , p) − ², and therefore v1 (X1 , q) ≤ v1 (X1 , p) − ² < v1 (B, p) − ² ≤ v1 (B, q), which contradicts the optimality of X1 .

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